Abstract
Indispensability arguments occupy a prominent role in discussions of mathematical realism. While different versions of these arguments are discussed in the literature, their general structure remains the same. These arguments contend that insofar as reference to mathematical objects is indispensable to science, we are committed to the existence of these ‘objects’. Unsurprisingly, much of the debate concerning indispensability arguments focuses on the crucial contention that mathematical objects are indispensable to science. For these arguments to provide support for mathematical realism, what we require are some compelling examples of mathematics playing an indispensable role in science. Towards this end, Alan Baker has supplied an example in which mathematics appears to be indispensable to a scientific explanation, what has come to be called the “Cicada MES”. This example has been the subject of much criticism, most notably that it depends on an unsupported empirical supposition. Recently, Baker proposed a number of revisions to the Cicada MES that aim to address some of these criticisms. In this paper, I argue that while Baker’s revisions go some way towards mitigating prominent criticisms of the Cicada MES, the underlying problem with the explanation remains. I argue that his revisions actually go in the wrong direction mathematically and empirically. In contrast, I propose a new Cicada MES. This new version accomplishes Baker’s goals and responds to the central objection to the explanation. It invokes a mathematical principle that plays a unifying role in the explanation, which makes providing adequate nominalist surrogates more difficult. And, more importantly, it accommodates an empirically grounded explanation of the phenomenon to be explained. Consequently, this new Cicada MES provides what proponents of indispensability arguments require, a compelling example of mathematics playing an indispensable role in science.
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Notes
This example was originally presented in Baker (2005).
Of course, this is not the only criticism that has been leveled against the Cicada MES. Some object that the mathematics in the example is only playing a representational role, not a genuinely explanatory role. See, for example, Saatsi (2011). Others object that the example is question begging. Even granting that mathematical objects are indispensable to explaining the behavior of Magicicadas, it seems that mathematical realism is presumed in the explanandum. What is to be explained, that Magicicadas have prime number periods, is not a purely physical explanandum. See Bangu (2008). Another objection, which applies to the Cicada MES, points to the arbitrariness of the chosen mathematical structure. The explanatory work could be done with sets and their properties (set theory) rather than numbers and their properties (number theory). Our choice to invoke the latter is arbitrary. This undermines the indispensability of the particular mathematical objects chosen, in this case numbers. (For a more detailed discussion of this ‘arbitrariness objection’, which is quite common in discussions of the role of mathematical entities in scientific explanation, see Field (1980, pp. 45–47). Field is not addressing this particular example, the Cicada MES, but the objection is the same.)
Baker (2016a, p. 336).
See Baron (2014) for another discussion of the Cicada MES as an optimization problem.
Baker (2016a, p. 336).
This slightly differs from the version of the principle that Baker presents. It includes the (relatively trivial) restriction that p is greater than 1. Without this restriction, the principle is false. If p = 1, then p is trivially coprime with every natural number less than it, but p is not prime.
Baker (2017, p. 781).
See Baker (2016a, p. 336). The sentiment that this is a ‘shaky’ component of the Cicada MES is reflected in the biological literature. When discussing the possibility of predators exerting pressures to explain the evolution of Magicicada periods, Lloyd and Dybas are careful to point out that “no synchronized predator appears to exist” (Lloyd and Dybas 1966a, p. 146). This concern is also expressed by mathematical biologist Horst Behncke who, when considering a mathematical model developed by Hoppensteadt and Keller to explore the hypothesis that predation could have driven Magicicada periods, contends that “the main disadvantage of [this] model is the fact that in nature there is no apparent predator with similar periodic activity” (Behncke 2000, p. 418). To make matters worse, when considering the possibility that the predators in question might have been parasitoid predators, Lloyd and Dybas note that this is to postulate “something which can never be verified directly” (Lloyd and Dybas 1966b, p. 500).
One might worry that original Cicada MES only generates the result that 13-years is optimal for Magicicada. What explains that some Magicicada have 17-year life cycles? To address this concern, proponents of the original Cicada MES can appeal to the hypothesis in Cox and Carleton (1988, 1998), which suggests that the existence of 13-year cicadas (together with their positing of periodical predators) explains the evolution of 17-year life cycles. (Put simply, not only do periodical predators negatively impact the insect’s fitness, interbreeding with cicada populations with unsynchronized cycles can negatively impact their fitness.) Note that a similar challenge is discussed in Section V. For a more detailed discussion of this issue, I refer the reader to this section. I mention this complication here merely to highlight that, even when we consider 17-year Magicicadas, it remains true that appealing to the original Cicada MES only requires positing periodical predators of every period less than 13-years. It explains the existence of 13-year Magicicadas, whose existence would, in turn, contribute to an explanation of the evolution of 17-year Magicicadas.
Baker (2017, p. 781).
One direction of this biconditional is trivial: If p is prime, then p is coprime with every prime less than p. This is a trivial consequence of P1. The other direction is slightly more interesting: If p is coprime with every prime less than p, then p is prime. As a quick proof, suppose that p is coprime with every prime less than p, but p is not prime. Then, p is product of primes, which are themselves less than p. Pick one of these primes, call it “q”. Since q is a factor of p and q is a prime less than p, this violates our supposition that p is coprime with every prime less than p.
It is worth noting that nothing prevents the proponent of the P3* Cicada MES from embracing the 4n + 1 Hypothesis. If, indeed, this hypothesis comes to be widely accepted among biologists, then proponents of the P3* Cicada MES ought to accept it. This would put the P3* Cicada MES on the same footing as the P2 Cicada MES when it comes addressing the question “Why are there no 19-year cicadas?” Of course, this would not address the underlying concerns discussed in this section. They would simply resurface with the question “Why are there no 29-year cicadas?”.
Baker (2017, p. 781).
It does not seem to me that the Cicada MES must explain why there are no 19-year cicadas; its aim is to explain why there are 13- and 17-year cicadas. An adequate explanation of the latter might (depending on the details) explain the former, but it does not seem that it must. To illustrate this point, suppose that a longshoreman arrives at work on Monday afternoon to discover four crates of fish on the dock. Upon querying the foreman about this phenomenon, the foreman explains that Bill (a local fisherman) had an excellent catch this morning. These crates came from his boat. While I suspect that we would readily accept this as a good explanation of the phenomenon, it does not address why there are not five crates, or six, or seven, or twenty, etc. Even though the foreman’s explanation does not address why there are not more crates, this does not seem to undermine our assessment of the quality of the explanation of the phenomenon that he is explaining (the presence of four crates). In fact, it would seem that explaining the absence of more crates requires a different kind of explanation—one that might invoke statistics concerning the common size of fishing hauls on Monday mornings, the number of fisherman who typically deliver to the dock, who (of these fisherman) were working that morning, etc.
Strictly speaking, one could eliminate the second interval, i.e., collapse its range to a single number, and still apply the principle. Were we to do this, however, it is no longer clear what the advantage would be in adopting P2 over the original P1.
See Koenig and Liebhold (2005).
I would like to thank an anonymous referee for drawing my attention to this benefit of the P3* Cicada MES.
P3* also enjoys greater flexibility in its potential applications. Note that P3* can accommodate evolutionary explanations that invoke P2. What P3* requires is that we ‘fill out’ the elements in P and E. The environmental constraints, which are required to specify the lower bound on the second interval in P2, supply the elements in E, and the elements in P are specified by the prime numbers in the first interval in P2. In other words, the data required to specify the intervals in P2 provide the necessary elements to employ P3*. However, while P3* can accommodate evolutionary explanations that employ P2, the converse is not always true. P2, which concerns a property of members of one interval of natural numbers with respect to members of a second interval, is more restrictive in its application.
To illustrate, consider the following scenario. Suppose we discover an organism that evolved life cycles of 9-years, and we discover that (a) there are recurring phenomena that adversely affected this organism’s populations every 2-, 5-, and 7-years (these could be peaks in predator populations, avoidance of interbreeding with other populations, or even periodical predators) and (b) there are environmental constraints that eliminated the viability of having a 3-year cycle. In this case, we can apply P3* to generate the desired optimality result, i.e., that, indeed, a 9-year cycle is optimal for this organism. Notice that P2 is not applicable in this scenario. The recurring adverse phenomena and environmental constraints do not provide the non-overlapping intervals that P2 requires. Put simply, the mathematics in P2 is unnecessarily restrictive. It includes artifacts (e.g., the requirement of non-intersecting intervals) that distract from what is relevant. In contrast, the principle P3* gets to the mathematical heart of the matter. It focuses directly on what is relevant—uncovering the elements in P and E that can drive the evolution of certain life cycles.
It might be argued that there is a sort of generality that Baker’s Cicada MES enjoys. In a recent article, Baker (2016b) shows that the explanation in the Cicada MES can be extended to explain optimal gear configurations for minimizing tire wear for brakeless fixed-gear bicycles. In particular, Baker shows that we can extend the reasoning in the Cicada MES to explain why a 14-tooth rear cog and a 47-tooth front cog are optimal for these bicycles. In his setup up, he takes the 14-tooth rear cog as a fixed parameter and considers what would be the optimal pairing when the range for the front cogs are cogs with 46-, 47-, 48-, and 49-teeth. As it turns out, the optimal pairing is (14, 47) since these are coprime and, consequently, minimize tire wear. (For the details, I refer the reader to Baker (2016b).) While this is an interesting extension of the reasoning in the Cicada MES, it is worth noting that both the P2 Cicada MES and the P3* Cicada MES can accommodate this explanation. To adopt P2 to this example, we need constraints (e.g., economic and manufacturing restrictions) that limit the viable range of front cog teeth to between 46- and 49-teeth. Among other things, such constraints remove from consideration any cogs of less than 46-teeth. To use P3* to generate Baker’s result, we only need to remove from consideration cogs of less than 46-teeth which do not have either 2 or 7 as factors. The factors of the 14-tooth rear cog (2 and 7) comprise the Ps in P3* and the cogs with less than 46-teeth that don’t have either 2 or 7 as factors comprise the Es. Consequently, while Baker’s extension of the Cicada MES to explain optimal cog configurations in brakeless fixed-gear bicycles is fascinating, the generality that it exemplifies is enjoyed by both the P2 Cicada MES and the P3* Cicada MES defended in this paper.
References
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Acknowledgements
I would like to thank the anonymous referees of this journal and the audience at the University of Delaware colloquium (who were presented an early version of this paper) for their helpful comments.
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Appendix
Appendix
P3*: Given two sets of primes P = {p1, …, pj} and E = {e1, …, em} such that P ∪ E comprises a complete set of primes {2, …, k} where k is greatest prime in P ∪ E, the least natural number n that is coprime with all pi (for 1 ≤ i ≤ j) and n ≠ ei (for 1 ≤ i ≤ m) is either es * es (where es is the smallest number in E that is not in P) or the least prime greater than k.
Proof
Suppose the background conditions are true. That is, we have two sets of primes, P and E, and they comprise a complete set of primes {2, … k} where k is the greatest prime in P ∪ E.
Suppose n is the least natural number such that n is coprime with all pi (for 1 ≤ i ≤ j) and n ≠ ei (for 1 ≤ i ≤ m)) but n is neither es * es nor the least prime greater than k. Clearly, the least prime greater than k (or, equivalently, the next prime in the series (2, …, k}) meets the requisite conditions. It is trivially true that such a prime would be coprime with all pi (for 1 ≤ i ≤ j) and not identical to any ei (for 1 ≤ i ≤ m)). Thus, since n is supposed to be the least natural number such that the aforementioned conditions hold, n must be less than the least prime greater than k. However, P ∪ E comprises a complete list of primes up to k. Accordingly, it cannot be that n is a prime less than the least prime greater than k and still meet the stipulated conditions. For, then, n would have to be identical to one of the primes in P or E, thereby violating either the coprimeness or inequality conditions, respectively. So, if n is to be less than the least prime greater than k and meet these conditions, n must be a composite. But one of the conditions on n is that it is coprime with all pi. As such, as a composite, n must be a product solely of the ei that are not in P. And, by hypothesis, n is not es * es. But, since es is the smallest ei not in E ∩ P, any other product of ei that are not in E ∩ P will clearly be greater than es * es, which contradicts our original assumption.
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Dieveney, P. Scientific explanation, unifying mathematics, and indispensability arguments (A new Cicada MES). Synthese 198, 57–77 (2021). https://doi.org/10.1007/s11229-018-01979-9
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DOI: https://doi.org/10.1007/s11229-018-01979-9